Degree of Freedomhttps://strathmaths.wordpress.com
Freedom is the freedom to say that two plus two make four.Fri, 06 Apr 2018 18:51:59 +0000enhourly1http://wordpress.com/https://s0.wp.com/i/buttonw-com.pngDegree of Freedomhttps://strathmaths.wordpress.com
A song for Pi Dayhttps://strathmaths.wordpress.com/2014/03/14/a-song-for-pi-day/
https://strathmaths.wordpress.com/2014/03/14/a-song-for-pi-day/#respondFri, 14 Mar 2014 07:00:15 +0000http://strathmaths.wordpress.com/?p=3005Continue reading →]]>If you’re reading this blog then you probably already know that today is Pi Day. In honour of the occasion, here’s a video in which our former student Chris Smith hymns the joys of — with the assistance of his Year 1 class and to what may be a rather familiar tune:

Chris is a maths teacher at Grange Academy in Kilmarnock. I’m grateful to him for permission to post this link and to Stephen Wilson bringing it to my attention!

(DP)

]]>https://strathmaths.wordpress.com/2014/03/14/a-song-for-pi-day/feed/0strathmathsEMS Popular Lecture: Maths and Sporthttps://strathmaths.wordpress.com/2014/03/04/ems-popular-lecture-maths-and-sport/
https://strathmaths.wordpress.com/2014/03/04/ems-popular-lecture-maths-and-sport/#respondTue, 04 Mar 2014 10:54:28 +0000http://strathmaths.wordpress.com/?p=3033Continue reading →]]>This year’s EMS Popular Lecture will take place on Friday 21 March in Edinburgh. Professor John Barrow will talk about “Maths and Sport”:

We will reveal some of the many ways in which mathematics helps us understand and improve sporting performance. Running, throwing, cycling, jumping, and weightlifting are among the examples we will take a look at from a new perspective. Along the way we will also see how Usain Bolt can break his world 100m record, investigate some odd scoring systems and see how delay differential equations help us understand American football.

(Personally, I doubt that even delay differential equations will ever explain American football to me, but I’m happy to give Prof. Barrow the benefit of the doubt…)

The lecture will take place at 4.30 p.m., in Lecture Theatre 3 of Appleton Tower, University of Edinburgh; tea will be served outside from 4.00 p.m. All are welcome, and there’s no charge for admission.

]]>https://strathmaths.wordpress.com/2014/03/04/ems-popular-lecture-maths-and-sport/feed/0strathmathsMaths makes your brain light up… but why?https://strathmaths.wordpress.com/2014/02/14/maths-makes-your-brain-light-up-but-why/
https://strathmaths.wordpress.com/2014/02/14/maths-makes-your-brain-light-up-but-why/#respondFri, 14 Feb 2014 15:25:33 +0000http://strathmaths.wordpress.com/?p=3024Continue reading →]]>By now you may have spotted the news story that the BBC headlined as “Mathematics: Why the brain sees maths as beauty”. As usual with science reporting, the headline doesn’t quite capture what the story’s about. The story’s based on a recently published paper by Zeki, Romaya, Benincase and Atiyah titled “The experience of mathematical beauty and its neural correlates”, and the key result of the study it describes is that when mathematicians look at an equation they regard as “beautiful”, there is activity in the same part of the brain that “lights up” when we look at other beautiful objects. Informally, we can conclude that mathematical beauty is experienced in a similar way to other forms of beauty — and perhaps that maths isn’t such an abnormal pleasure after all…

The famous dead salmon experiment should make us sceptical about reading too much into neuroimaging, but in justice to these researchers they seem to have been very careful in their analysis and they’re also careful not to overstate what they claim. Two aspects of the work, apart from the headline result, might be of particular interest to mathematicians. One is the question of which equations were regarded as particularly beautiful; the other is the relationship between understanding an equation and appreciating its beauty.

The complete list of equations used in the experiments is available as “Data Sheet 1” from the sidebar of the online article, and their scores in the pre-experiment beauty contest are in the spreadsheet “Data Sheet 3”. (You mght like to take a look at these and see how you rate them…) The top-rated equation was an old classic, Euler’s identity

.

Also jostling for positions on the podium were the Pythagorean identity

I found myself less surprised by the top of the league table than by the bottom of it: Euler’s identity regularly tops lists of the most beautiful equations out there, while it as well as Pythagoras’s theorem and the complex exponential connect elementary mathematical objects in a very fundamental (and useful) way. The beauty of the Cauchy–Riemann equations is perhaps less apparent until you’ve done a first course in complex analysis and come to realise just how strongly they structure the behaviour of analytic functions.

At the bottom, I guess Euler’s polyhedron formula just looks too simple to tickle many people’s fancy — which is a shame for the philosophers of mathematics, because the derivation of this formula is a classic (and much-debated) case study in how mathematical results are obtained. I was surprised to see Ramanujan’s formula down there, because for me such expressions pack a considerable punch due to their sheer unexpectedness… but it’s possible that Ramanujan’s “strange Indian melodies”, as Douglas Hofstadter called them, strike the brain rather more like one of Courtney Pine’s further-out solos than one of Mozart’s sonatas — that is to say, our initial reaction is puzzlement, followed by appreciation, rather than pure enjoyment from the outset.

The Gauss–Bonnet and spectral theorems, I suspect, suffered from a different handicap, which is that they are sufficiently “technical” that many mathematicians (myself included) simply don’t know their context well enough to appreciate them. This brings us to the second interesting aspect of the study: it seems that to appreciate mathematical beauty it really helps if we understand what we’re looking at.

When non-mathematicians were shown the equations, they identified some of them (though not many!) as more beautiful than others, but their judgement didn’t coincide closely with that of the mathematicians — the researchers conclude that they “did so on the basis of the formal qualities of the equations”, or to put it more colloquially, whether they made pretty shapes on the page. For mathematicians, on the other hand, there was a significant correlation between their understanding of an equation and how beautiful they rated it as being. The intellectual beauty that we experience in mathematics really does seem to be earned by our knowledge of mathematics — to borrow a line from Plato, quoted by the researchers, “nothing without understanding would ever be more beauteous than with understanding”.

Nevertheless, the correlation between beauty and understanding wasn’t perfect. Our experience of mathematical beauty, it seems, is more than just a sense of satisfaction that we’ve understood something: understanding, perhaps, acts only as a gateway to appreciation, and exactly what else contributes to that appreciation lies beyond the ken of neuroimaging. It seems fitting that the paper ends, like all honest discussions of aesthetics, on an open note:

… that there was an imperfect correlation between understanding and the experience of beauty and that activity in the mOFC [the relevant bit of the brain] cannot be accounted for by understanding but by the experience of beauty alone, raises issues of profound interest for the future. It leads to the capital question of whether beauty, even in so abstract an area as mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved…

Hence the work we report here, as well as our previous work, highlights further the extent to which even future mathematical formulations may, by being based on beauty, reveal something about our brain on the one hand, and about the extent to which our brain organization reveals something about our universe on the other.

And possibly that is all we can know for the moment, even if it’s not quite all we need to know…

(DP)

]]>https://strathmaths.wordpress.com/2014/02/14/maths-makes-your-brain-light-up-but-why/feed/0strathmathsDetail from the Sosibios vase (Keats's original Grecian Urn)RSS event: statistics and the referendumhttps://strathmaths.wordpress.com/2014/02/07/rss-event-statistics-and-the-referendum/
https://strathmaths.wordpress.com/2014/02/07/rss-event-statistics-and-the-referendum/#respondFri, 07 Feb 2014 15:35:19 +0000http://strathmaths.wordpress.com/?p=3021Continue reading →]]>Here at DoF we try to stay away from politics, or at least from the sort of politics that might get us accused of lobbying. However, even we have noticed that a vote is due to take place in September that might be of some importance for our wee country (or our wee corner of a bigger country, if you prefer). So have the Royal Statistical Society, and they’re putting on an event at the Parliament in Edinburgh on the afternoon of March 26. It will look at statistical issues brought into focus by the independence debate, including those concerning the economy and health as well as the opinion polls themselves. It’s open to all, but registration (via the RSS Glasgow events page) is essential.

(DP)

]]>https://strathmaths.wordpress.com/2014/02/07/rss-event-statistics-and-the-referendum/feed/0strathmathsTwo forthcoming IMA seminarshttps://strathmaths.wordpress.com/2014/02/06/two-forthcoming-ima-seminars/
https://strathmaths.wordpress.com/2014/02/06/two-forthcoming-ima-seminars/#respondThu, 06 Feb 2014 17:07:44 +0000http://strathmaths.wordpress.com/?p=3017Continue reading →]]>If you’re interested in applications of mathematics, you might want to check out two upcoming IMA seminars, one in Edinburgh and one in Glasgow. On 20 February in Edinburgh, Prof. Murray Campbell will talk on “Applications of Mathematics in understanding the creation and perception of musical sounds”, and on 11 March in Glasgow, Dr Andrew Fletcher will talk on “The Mathematics of Astronomy”. Both talks are open to all comers, whether IMA members or not. More details are available on the IMA website.

(DP)

]]>https://strathmaths.wordpress.com/2014/02/06/two-forthcoming-ima-seminars/feed/0strathmathsWit and heuristics in crosswords and mathematicshttps://strathmaths.wordpress.com/2014/01/28/wit-and-heuristics-in-crosswords-and-mathematics/
https://strathmaths.wordpress.com/2014/01/28/wit-and-heuristics-in-crosswords-and-mathematics/#respondTue, 28 Jan 2014 14:40:46 +0000http://strathmaths.wordpress.com/?p=2990Continue reading →]]>In a recent post on the SIAM blog, Professor Des Higham discusses cryptic crosswords and the reasons why they appeal to so many mathematicians of his acquaintance. I’m one of those mathematicians, having tried my hand both at solving and at compiling crosswords, and Des’s article got me thinking a little more about where their appeal really lies. Clearly there’s the fact that mathematicians, in general, simply like solving puzzles; indeed, the boundaries between recreational puzzles and “genuine” mathematics can sometimes become so blurred as to be non-existent, as in the early work on gambling or the Königsberg bridges problem. But can we, I wondered, go a bit further than this and look at crossword-solving through the lens of mathematical thinking, or vice versa?

Araucaria araucana, the monkey puzzle tree. [Photo: Beth Hemmila.] “A mathematician who can only generalise is like a monkey who can only climb UP a tree… and a mathematician who can only specialise is like a monkey who can only climb DOWN a tree” (George Polya).

I think that one of Des’s statements, which is that “solutions are discovered through creative use of logical steps”, requires a little expansion. In my experience — which might be entirely atypical — the steps involved in discovering a solution are very far from logical: they involve vague associations, working hypotheses that are briefly considered and then discarded, and occasional outrageous guesses; and it is only after a solution has presented itself that formal reasoning steps in to check whether the solution is in fact valid. In this respect, of course, crosswords may have still more in common with mathematical research than Des suggests!

Perhaps the best way to express this is to say that what crossword solving shares with mathematics is the eerie power of a good heuristic. This shows up most obviously in anagrams. By my reckoning, there are 9! = 362880 ways to arrange the letters in a nine-letter anagram (not at all uncommon in an everyday grid) and 15! = 1307674368000 ways to arrange the letters in a fifteen-letter anagram (rarer but hardly unknown). Despite this, many solvers have had the experience of looking at a lengthy anagram and spotting the answer within a second or so. It seems barely plausible that the brain has had time to implement a brute-force strategy such as working through all the possible combinations of letters, or even (more plausibly) thumbing through a mental dictionary of nine- or fifteen-letter words and checking each for compliance with the clue. Rather, what seems to happen is that we spot certain combinations of letters (like a T, an I, an O and an N), and consider the working hypothesis that our solution ends in -TION: this gives us a massively reduced brute-force problem to solve, and because of the highly restricted set of letter combinations to be found in any natural language, the strategy works pretty frequently. (The hardest anagrams to spot tend to be those of words which contain unusual letter combinations and don’t contain common prefixes or suffices — the well-known anagram of CARTHORSE is a case in point.)

Paradoxically, the longer an anagram becomes the easier it can be to solve: with some of the late and irreplaceable Araucaria’s colossal quotation anagrams, the most successful solution strategy was simply to look at the pattern of letters and wait for a plausible sequence of words to present itself. (To take a very sub-Araucarian example, if the pattern of letters is (2, 2, 2, 3, 2, 2) and you have a suspicion that the crossword is themed around lines from Shakespeare, then a possibility will rapidly present itself and render the clue “i.e.?” comprehensible!) In a cryptic clue, of course, the problem is rendered even easier by the promise that the clue will contain, somewhere, a “straight” definition of the solution, so another effective strategy is to consider the possible definitions in turn, and for each one riffle through your mental thesaurus looking for plausible candidates for the anagram.

The reason why heuristics work, in crosswords as in mathematics, is that we are playing a game in a highly conventionalised world. This is obvious in mathematics, where there are formal rules that determine whether a particular step is admissible or not, and where, less formally, our developing mathematical maturity gives us a sense of which claims are credible and which methods could plausibly lead to particular results. (A nice illustration of these informal rules is given by Scott Aaronson’s Ten signs a claimed mathematical breakthrough is wrong.) Crosswords, less obviously, don’t just work within a restricted “language” (there are, after all, only so many ways to indicate that a clue contains an anagram) and a set of formal rules (usually the Ximenean rules); they also have a surprisingly restricted field of reference. It’s hard, in my experience, to score a reasonable percentage against the broadsheet prize crosswords without some vague knowledge of classical music and cricket (both of which have substantial vocabularies crammed with specialised and quite odd words), but I can testify that it’s perfectly possible to get by with very little knowledge of, say, association football or popular music recorded since about 1995. In a sense — and I don’t want to push the analogy too far — the odd areas of knowledge that help disproportionately with crosswords are a bit like the collection of mathematical techniques and tricks that each of us learns are disproportionately helpful solving problems in our own area of mathematics. (An example of such a trick from my own field, fluid dynamics, is that of looking for self-similar solutions; I’m sure my colleagues in other disciplines have their own favourites.)

Of course, the danger is that, without an injection of the unexpected, crosswords can become rather stereotyped, and I think the same applies to mathematics. A great compiler has what I think is best described as wit: the ability to startle the solver with a surprisingly topical reference or linguistic oddity (the famous anagram of BRITNEY SPEARS, though it’s a bit dated now, is a case in point), or simply to approach the task of clueing from an entirely unexpected angle — again, Araucaria provides some perfect examples. Similarly, the difference between a journeyman piece of mathematics and the work of a real master is that the former will tackle a standard problem very competently in a standard way, whereas the latter will deploy an idea or a technique from an unexpected source, or address a problem that nobody beforehand would have thought was natural or feasible to tackle.

That is the lasting joy of both crosswords and mathematics: not bypassing the heuristics and the conventional (or even algorithmic) elements, but mastering them enough to appreciate the wit that operates within that framework and occasionally dances beyond it. It’s the joy that one also finds in poetry and in music and, I suspect, in every other sphere where an apparently restrictive system of rules becomes the essential support for creativity.

(DP)

PS: ORCHESTRA; TO BE OR NOT TO BE; PRESBYTERIANS. There: you can stop worrying now…

]]>https://strathmaths.wordpress.com/2014/01/28/wit-and-heuristics-in-crosswords-and-mathematics/feed/0strathmathsMonkey puzzle treeChain reaction forceshttps://strathmaths.wordpress.com/2014/01/15/chain-reaction-forces/
https://strathmaths.wordpress.com/2014/01/15/chain-reaction-forces/#respondWed, 15 Jan 2014 13:42:55 +0000http://strathmaths.wordpress.com/?p=2981Continue reading →]]>Here’s a weird little phenomenon that should appeal to all of us who think we understand Newtonian mechanics…

First, have a look at this 36-second Youtube video from the excellent Steve Mould:

Now try to work out what on earth is going on… Bear in mind that this behaviour doesn’t rely crucially on the beads along the chain, and you can even get the same effect with rope:

If you’re puzzled, you’re not the only one. Steve Mould drew attention to the phenomenon in a blog post back in the summer (it also contains a link to a really neat slow-motion video), which in turn produced some discussion on Reddit (and some follow-up experiments).

Most recently, an analysis by two professional physicists, Prof. Mark Warner and Dr John Biggins, has appeared in the journal Proceedings of the Royal Society A. [You should be able to read the full article if you’re on a Strathclyde computer; if not then you might be interested in the podcast version.] If their analysis is correct, then all you need to know in order to understand the chain fountain is some school-level physics (tension, gravity, and a little bit of moments) — and a willingness to think carefully in some slightly non-intuitive directions. (Spoiler: it turns out that the really crucial detail is the reaction force that the jar exerts on the chain as it’s picked up. But you’d already guessed that… right?)

(DP)

]]>https://strathmaths.wordpress.com/2014/01/15/chain-reaction-forces/feed/0strathmathsRSS seminar: statistics making an impacthttps://strathmaths.wordpress.com/2014/01/06/rss-seminar-statistics-making-an-impact/
https://strathmaths.wordpress.com/2014/01/06/rss-seminar-statistics-making-an-impact/#respondMon, 06 Jan 2014 14:29:37 +0000http://strathmaths.wordpress.com/?p=2974Continue reading →]]>John Pullinger, the president of the Royal Statistical Society, will be giving a seminar in the Department on Friday 31 January. He has a lengthy career in the civil service as a statistician and currently works at the House of Commons, so it should be an interesting insight into the use of statistics in and by government! According to the advert, the speaker will “explore how the role of statistics in society has changed with changes in the politics of decision-making. He will outline the Royal Statistical Society strategy for the next 4 years and consider how individual members of the RSS can play a part.”

The event will take place from 1500-1600 in LT908, Livingstone Tower, on Friday 31 January, and will be followed by a wine reception. It’s free and open to all, and non-members are very welcome!

(DP)

]]>https://strathmaths.wordpress.com/2014/01/06/rss-seminar-statistics-making-an-impact/feed/0strathmathsQuotation for the new yearhttps://strathmaths.wordpress.com/2014/01/01/quotation-for-the-new-year/
https://strathmaths.wordpress.com/2014/01/01/quotation-for-the-new-year/#respondWed, 01 Jan 2014 07:00:17 +0000http://strathmaths.wordpress.com/?p=2968Continue reading →]]>For anyone who is currently contemplating their January exam revision through bleary eyes and wondering what the point of their maths degree was supposed to be, here’s a possible answer from the lawyer and philosopher Francis Bacon:

In the mathematics I can report no deficience, except it be that men do not sufficiently understand this excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be too dull, they sharpen it; if too wandering, they fix it; if too inherent in the sense, they abstract it. So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures, so in the mathematics that use which is collateral and intervenient is no less worthy than that which is principal and intended.

Happy New Year! Have fun, but I hope your “wit and faculties intellectual” are thoroughly remedied by the time you return to campus…

(DP)

]]>https://strathmaths.wordpress.com/2014/01/01/quotation-for-the-new-year/feed/0strathmathsLinks: more on the mathematics of musichttps://strathmaths.wordpress.com/2013/11/28/links-more-on-the-mathematics-of-music/
https://strathmaths.wordpress.com/2013/11/28/links-more-on-the-mathematics-of-music/#commentsThu, 28 Nov 2013 09:25:35 +0000http://strathmaths.wordpress.com/?p=2966Continue reading →]]>For those of you who read our earlier article on the mathematics behind music and want to know more, here are a couple of links that might well interest you.

Amazingly mathematical music is an online article from Washington University in St Louis, which explores some of the mathematical patterns to be found in both musical harmonies and rhythms. It’s well illustrated (if that’s the word) with sound clips including the infamous perpetually ascending stairs illusion, barbershop choruses and Tuvan throat singing — well worth checking out! It’s accompanied by a podcast interview with David Wright, who is both the chair of the Mathematics Department at Washington University and a very successful a capella choir director.